Question

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. tan x(cot x - cos x)​

Answer 1

Answer: Starting with the expression:

tan(x)(cot(x) - cos(x))

Recall that cot(x) is the reciprocal of tan(x), so we can substitute cot(x) = 1/tan(x):

tan(x)(1/tan(x) - cos(x))

Simplifying:

1 - cos(x)tan(x)

Next, we can use the identity cos(x) = 1/sec(x) to eliminate the cotangent term:

1 - (1/sec(x))tan(x)

Now, we can use the identity tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x) to express the expression in terms of sine and cosine:

1 - (1/cos(x))(sin(x)/cos(x))

Simplifying:

cos(x)/cos(x) - sin(x)/cos(x)

= (cos(x) - sin(x))/cos(x)

Therefore, the simplified expression in terms of sine and cosine is:

(cos(x) - sin(x))/cos(x)

Explanation: